<< Chapter < Page Chapter >> Page >

Key equations

quadratic formula x = b ± b 2 4 a c 2 a

Key concepts

  • Many quadratic equations can be solved by factoring when the equation has a leading coefficient of 1 or if the equation is a difference of squares. The zero-factor property is then used to find solutions. See [link] , [link] , and [link] .
  • Many quadratic equations with a leading coefficient other than 1 can be solved by factoring using the grouping method. See [link] and [link] .
  • Another method for solving quadratics is the square root property. The variable is squared. We isolate the squared term and take the square root of both sides of the equation. The solution will yield a positive and negative solution. See [link] and [link] .
  • Completing the square is a method of solving quadratic equations when the equation cannot be factored. See [link] .
  • A highly dependable method for solving quadratic equations is the quadratic formula, based on the coefficients and the constant term in the equation. See [link] .
  • The discriminant is used to indicate the nature of the roots that the quadratic equation will yield: real or complex, rational or irrational, and how many of each. See [link] .
  • The Pythagorean Theorem, among the most famous theorems in history, is used to solve right-triangle problems and has applications in numerous fields. Solving for the length of one side of a right triangle requires solving a quadratic equation. See [link] .

Section exercises

Verbal

How do we recognize when an equation is quadratic?

It is a second-degree equation (the highest variable exponent is 2).

Got questions? Get instant answers now!

When we solve a quadratic equation, how many solutions should we always start out seeking? Explain why when solving a quadratic equation in the form a x 2 + b x + c = 0 we may graph the equation y = a x 2 + b x + c and have no zeroes ( x -intercepts).

Got questions? Get instant answers now!

When we solve a quadratic equation by factoring, why do we move all terms to one side, having zero on the other side?

We want to take advantage of the zero property of multiplication in the fact that if a b = 0 then it must follow that each factor separately offers a solution to the product being zero: a = 0   o r  b = 0.

Got questions? Get instant answers now!

In the quadratic formula, what is the name of the expression under the radical sign b 2 4 a c , and how does it determine the number of and nature of our solutions?

Got questions? Get instant answers now!

Describe two scenarios where using the square root property to solve a quadratic equation would be the most efficient method.

One, when no linear term is present (no x term), such as x 2 = 16. Two, when the equation is already in the form ( a x + b ) 2 = d .

Got questions? Get instant answers now!

Algebraic

For the following exercises, solve the quadratic equation by factoring.

x 2 9 x + 18 = 0

x = 6 , x = 3

Got questions? Get instant answers now!

6 x 2 + 17 x + 5 = 0

x = 5 2 , x = 1 3

Got questions? Get instant answers now!

3 x 2 75 = 0

x = 5 , x = −5

Got questions? Get instant answers now!

4 x 2 = 9

x = 3 2 , x = 3 2

Got questions? Get instant answers now!

5 x 2 = 5 x + 30

x = −2 ,

Got questions? Get instant answers now!

7 x 2 + 3 x = 0

x = 0 , x = 3 7

Got questions? Get instant answers now!

For the following exercises, solve the quadratic equation by using the square root property.

x 2 = 36

x = −6 , x = 6

Got questions? Get instant answers now!

( x 1 ) 2 = 25

x = 6 , x = −4

Got questions? Get instant answers now!

( 2 x + 1 ) 2 = 9

x = 1 , x = −2

Got questions? Get instant answers now!

For the following exercises, solve the quadratic equation by completing the square. Show each step.

x 2 9 x 22 = 0

x = −2 , x = 11

Got questions? Get instant answers now!

x 2 6 x = 13

x = 3 ± 22

Got questions? Get instant answers now!

2 + z = 6 z 2

z = 2 3 , z = 1 2

Got questions? Get instant answers now!

2 x 2 3 x 1 = 0

x = 3 ± 17 4

Got questions? Get instant answers now!

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 7

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask